Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with Delay and Dependent Risks

This paper investigates a robust optimal excess-of-loss reinsurance and investment problem with delay and dependent risks for an ambiguity-averse insurer (AAI).TheAAI’s wealth process is assumed to be two dependent classes of insurance business. He/she can purchase excess-of-loss reinsurance from the reinsurer and invest in a risk-free asset and a risky asset whose price follows Heston model. We obtain the explicit expressions of the optimal excess-of-loss reinsurance and investment strategy by maximizing the expected exponential utility of AAI’s terminal wealth. Finally, we give the proof of the verification theorem. Our models and results posed here can be regarded as a generalization of the existing results in the literature.


Introduction
In the past decade, the topic about optimal investment and reinsurance problems has attracted a lot of attention. These optimal problems have been studied in terms of various objectives, for example, [1][2][3][4] considered the objective function of minimizing the ruin probability; [5][6][7][8][9] studied the optimal problems aiming to maximize the survival probability or the expected utility of terminal wealth; [10][11][12][13] investigated optimal reinsurance and investment problems under mean variance criterion.
Although so many notable scholars have considered the optimal reinsurance and investment problems, two important aspects are still being worthy of further exploration. One is lack of considering ambiguity, and the other one is the optimal control problems under delayed systems. On one hand, the model uncertainties do exist widely in finance, especially in insurance, the field of asset pricing, consumption, and portfolio selection. As a result, the ambiguity-averse insurer (AAI) has to look for a methodology to handle this uncertainty. One possible way is to use the robust approach, where some alternative models closed to the estimated model are introduced and the robust optimal strategy is obtained. Recently, some scholars paid more attention to optimal investment-reinsurance problems with ambiguity. Reference [14] assumed that the insurer's wealth process follows a diffusion model, and they optimized a proportional reinsurance and investment problem with model uncertainty. Reference [15] obtained the robust optimal proportional reinsurance and investment strategies for an AAI; in their article, the surplus process is assumed be a Cramér-Lundberg risk model and the risky asset's price follows a constant elasticity of variance (CEV) model. Reference [16] studied the robust optimal proportional reinsurance and investment strategies for both an insurer and a reinsurer. Reference [17] took default risk into account and derived the robust optimal control strategy under variance premium. Different from the above-mentioned literature, [18] analyzed a robust optimal problem of excess-of-loss reinsurance and investment in a model with jumps for an AAI.
On the other hand, the investors or the insurers make important future decisions only according to the present states of a system, but they do not consider the past states. However, the future states of a system usually may depend on its past states, which do exist in our real-world systems. For example, in the stock market, investors not only are concerned with the present stock price but also pay more attention to the trend of the stock price in the past periods. Thus, it is more realistic to take some past information of the system into account. Due to the structure of infinitedimensional state space, generally speaking, it is difficult to solve these stochastic control problems with delay analytically. As a result, there is an explicit solution for this problem. Only those problems, in which some special forms of delay information are considered in the state process, are found to be finite-dimensional and then can be solved (see, for example, [19][20][21]). Moreover, the delay is first introduced into the optimal proportional reinsurance and investment problems by [22] under mean-variance criterion. Reference [23] optimized the delayed problem of excessof-loss reinsurance and investment under maximizing the expected exponential utility of the insurer's terminal wealth. Reference [24] took multiple dependent classes of insurance business into consideration, investigated the time-consistent reinsurance-investment problem with delay and derived the optimal strategy under the mean-variance criterion. As mentioned in [24], in fact, some insurance businesses are usually correlated by some way in practice. For example, a traffic accident (or fire accidents or car accidents or aviation accidents and so on) may cause property loss or medical claims or death claims; these insurance businesses will be correlated. Therefore, it is necessary to take dependent risks into account in the actuarial literature. References [25,26] assumed that the insurer's surplus process consists of two or more dependent classes of insurance business and the claim number processes are correlated through a common shock component, and they discussed optimal proportional reinsurance problems under the criterion of maximizing the expected utility of terminal wealth. For other research about dependent risks, we refer the readers to [27][28][29][30][31] and the references therein. This paper takes excess-of-loss reinsurance into account, which is better than proportional reinsurance in most situations; see [32]. Suppose that the insurer's wealth process consists of two dependent classes of insurance business. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market which consists of a risk-free asset and a risky asset. The risky asset's price is described by Heston model. Moreover, it is assumed that there exists capital inflow into or outflow from the insurer's current wealth. Given that the insurer's claim process and the risky asset price (true model) may deviate from a relative good estimated model (reference model) in real-word, the model uncertainty should be taken into consideration. On the basis of the above setup, we first formulate a robust optimal control problem with delay and dependent risks and then investigate the optimal strategy for an AAI by maximizing the expected exponential utility of terminal wealth. This paper has the following main contributions: (i) an optimal excess-of-loss reinsurance and investment problem with dependent risks is studied; (ii) both ambiguity and the capital inflow/outflow are introduced into this problem; (iii) some special cases are provided, such as the case of investmentonly, ambiguity-neutral insurer, and no delay, which demonstrates that our model and results can be considered as a generalization of the existing results in some literature, e.g., [23,25].
The rest of this paper is structured as follows. We present the formulation of our model in Section 2. Section 3 discusses the robust optimal strategy and derives the optimal results. Section 4 is devoted to proving the verification theorem. Some special cases of our model are provided in Section 5. Section 6 concludes the paper. In Appendix, technical proofs are presented.

Model Formulation
We consider a filtered complete probability space (Ω, F, {F } ∈[0, ] , ), where represents the terminal time and is a positive finite constant and F stands for the information of the market available up to time . Assume that all processes introduced below are well-defined and adapted processes in this space. In addition, suppose that trading takes place continuously and involves no taxes or transaction costs and that all securities are infinitely divisible.

Surplus Process.
This section presents a risk model consisting of two dependent classes of insurance business. The insurer's wealth process is modeled of where the positive constant is the premium rate; is the ith claim size from the first class; { , ≥ 1} are assumed to be i.i.d. positive random variables with common distribution function (⋅), finite first moment ( ) = > 0, and second moment ( 2 ) = 2 ; is the ith claim size from the second class and { , ≥ 1} are assumed to be i.i.d. positive random variables with common distribution function (⋅), finite first moment ( ) = > 0, and second moment ( 2 ) = 2 ; { 1 ( ), ≥ 0}, { 2 ( ), ≥ 0} and { ( ), ≥ 0} are three independent Poisson processes with positive intensity parameters 1 , 2 , and , respectively.
where > 0 is the insurer's safety loading from the ith claim.

Excess-of-Loss
Reinsurance. Suppose that the insurer can purchase excess-of-loss reinsurance by reducing the Discrete Dynamics in Nature and Society 3 underlying claims risk. Denote by 1 and 2 the excess-ofloss retention levels, and let be the parts of the first claims and the second claims held by the insurer, respectively. Then by (1), the wealth process becomes with the premium rate where is the reinsurer's safety loading from the ith claim. Assume that > , which implies that the reinsurance is not cheap. According to [33], the wealth process (4) can be approximated by the following diffusion model: ( ) and ( ) are two standard Brownian motions whose correlation coefficient is and 0 ( ) is another standard Brownian motion dependent of ( ) and ( ).

Financial Market.
The insurer is assumed to invest in a risk-free asset whose price process ( ) is governed by and a risky asset whose price process ( ) follows Heston model, where positive constant is the risk-free interest rate, , , , and are all positive constants, and 1 ( ) and 2 ( ) are two standard Brownian motions with [ 1 ( ) 2 ( )] = , ∈ [−1, 1]. By standard Gaussian linear regression, 2 ( ) can be rewritten as where 2 ( ) is another standard Brownian motion. We assume that 0 ( ), 1 ( ), and 2 ( ) are mutually independent. Moreover, we require 2 ≥ 2 to ensure that ( ) is almost surely nonnegative. means "no reinsurance" and ( ) = 0 means "full reinsurance," and ( ) is the money amount invested in the risky asset at time t, the amount of money invested in the risk-free asset at time t is ( ) − ( ), and here ( ) is the insurer's wealth after adopting strategy . Thus, the evolution of ( ) is governed by It is noted that the wealth process is traditionally formulated as (14), which is a stochastic differential equation (SDE) without delay. In the sections below, we will formulate a wealth process with delay, which is caused by the instantaneous capital inflow into or outflow from the insurer's current wealth. The delayed wealth process is still denoted by (⋅). Let ( ), ( ), and ( ) be the delayed wealth and average and pointwise performance of the wealth in the past horizon [ − ℎ, ], respectively, i.e.
for ∀ ∈ [0, ], where ≥ 0 is an average parameter and h > 0 is the delay parameter. Denote by the function the capital inflow/outflow amount, where ( ) − ( ) represents the absolute performance of wealth between t and − ℎ, and ( ) − ( ) stands for the average performance of the wealth in [ −ℎ, ]. Such capital inflow or outflow, which is related to the past performance of the wealth, may come out in various situations. For example, a good past performance of the wealth may bring the insurer more gain. On the contrary, a poor past performance of the wealth may force the insurer to seek further capital injection to cover the loss so as to achieve the final performance objective. Following [22,23], when we consider such a capital inflow/outflow function, the wealth process ( ) can be given as follows: Such capital inflow/outflow, which is related to the past performance of the wealth, may come out in various situations. For example, a good past performance of the wealth may bring the insurer more gain and further the insurer can pay a part of the gain as dividend to his/her stakeholders, in this case < 0. On the contrary, a poor past performance of the wealth may force the insurer to seek further capital injection to cover the loss so as to achieve the final performance objective and in this case > 0. To make this problem solvable, we assume that the amount of the capital inflow/outflow is proportional to the past performance of the insurer's wealth, i.e.
where 1 and 2 are two nonnegative constants. Inserting (6), (11), (12), and (19) into (18) leads to the following stochastic differential delay equation (SDDE): where Furthermore, assume that ( ) = 0 > 0, ∀ ∈ [−ℎ, 0], which means that the insurer is endowed with the initial wealth 0 at time −ℎ and does not start the investment and (re)insurance business until time 0. Therefore, the initial value of the average performance wealth (0) is Given that the investment performance has an effect on the insurer's wealth, this paper assumes that the insurer is concerned with ( ) and ( ) in the time interval [ − ℎ, ]. Moreover, suppose that the insurer has the following exponential utility function defined by where V > 0 and 0 < < 1 are constants. Here V represents a constant absolute risk aversion coefficient which plays a vital role in insurance practice and actuarial mathematics. Note that is the weight of ( ), so ( ) will impact the average performance of the terminal wealth. We consider the integrated delayed wealth ( ) rather than the average one ( ) directly. Define = / ∫ 0 −ℎ d as the transformed weight, which can be considered as the weight between ( ) and ( ). Thus (23) is equivalent to considering For simplicity, we call the combination ( ) + ( ) the terminal wealth. In fact, our modeling framework for the term ( ) + ( ) is consistent with the classical literature (e.g., [34,35]) on stochastic control problem with delay. Reference [23] considered a utility function, which is similar to (23) in our paper and studied an optimal problem with delay for an insurer; in their article, the insurer's surplus process is assumed to follow the classical Cramér-Lundberg model. Compared with [23], we not only incorporate model uncertainty into our study which will be introduced later but also assume that the wealth process consists of two classes Discrete Dynamics in Nature and Society 5 of insurance business, in which the two claim processes are dependent.
In traditional, it is assumed that the insurer is ambiguityneutral with the following objective function: where (⋅) is the expectation under the probability measure P andΠ is the set of admissible strategies which will be defined in Definition 1. However, many insurers are ambiguity-averse and always try to guard themselves against worse-case scenarios. Thus, it is reasonable to suppose an insurer is ambiguity-averse in the field of insurance. In what follows, we present a robust portfolio choice with uncertainty for an AAI. Suppose that the AAI has a relative good estimated model (also called reference model) to describe the risky assets prices and his/her claim process, but he/she is always skeptical about this reference model and hopes to take alternative models into account. According to [36], the alternative models are defined by a set of probability measures Q which are equivalent to the as follows: Definition 1. For any fixed ∈ [0, ], the strategy is said to be admissible if it is F -progressively measurable and satisfies Next, define a process For ∀ ∈ Θ, we define a real-valued process where ( ) = ( 0 ( ), 1 ( ), 2 ( )) . By Ito's differentiation rule, Thus Λ ( ) is a P-martingale. Hence, [Λ ( )] = 1. For ∀ ∈ Θ, a new real-word probability measure absolutely continuous to on F is defined by So far, we have constructed a family of real-world probability measures parameterized by ∈ Θ. Applying Girsanov's theorem, we can see that under the alternative measure , is a standard threedimension Brownian motion, where Note that the alternative models in class Q only differ in the drift terms. Thus, the risky asset's price (12) under is And the wealth process (20) under is rewritten as Inspired by [37,38], we formulate the following robust control problem to modify problem (25), i.e., where Ψ ( , ( ) , ( ) , ( )) = ‖ ( )‖ 2 2 ( , ( ) , ( )) , (35) and , , , is calculated under . In (34), the deviations from the reference model are penalized by the second term in the expectation. In fact, this penalty term depends on the relative entropy arising from diffusion risk. In addition, the parameter in (35) represents the strength of the preference for robustness. For analytical tractability, suppose that the parameter in (35) is state-dependent. In particular, following [37,39], we set where (≥ 0) stands for the ambiguity-aversion coefficient and describes the AAI's attitude to the diffusion risk.
For convenience, we first provide some notations. Let To make the problem (34) solvable, by dynamic programming principle, the robust Hamilton-Jacobi-Bellman (HJB) equation for (34) can be derived as (see [39,40]): for < with boundary condition where A , is the generator of (33) under and is defined as Here, is a short notation for ( , , , )

Robust Optimal Results with Delay
The aim of this section is to find the robust optimal control strategy for problem (34) under the exponential utility. As mentioned above, in general, the delayed control problem is infinite-dimensional. In order to make this problem be finitedimensional and solvable, according to [23], we assume the parameters satisfies the following conditions: It is noted that the above two conditions are the sufficient conditions for the optimal control problem with delay, which guarantee that the HJB equation (37) has a closed-form solution. Furthermore, they help us explore the implication of the problems with delay and without delay.
Step 3. Inserting ( * 1 , * 2 ) into (49) yields According to the arbitrariness of and , (55) is equivalent to the two following equations: For (56), taking the boundary condition ( ) = 1 into account yields In order to determine the point ( * 1 , * 2 ) clearly, (52) is transformed into or equivalently, Next, we need to define three following auxiliary functions: For convenience, we assume that 1 ≥ 2 . It is easy to verify that both ( ) and ( ) are strictly increasing functions for ≥ 0, so their inverse functions −1 ( ) and −1 ( ) exist. From (60), we get ( 1 ) = ( 2 ), and then Plugging (62) and (58) into the second equation of (52), we obtain Therefore, let If the equation ℎ( ) = 0 has a solution on [0, 1 ], the solution is indeed * 1 we try to derive, and as a result, * 2 will be easily determined. Then, the robust optimal reinsurance strategy can be derived and summarized in the following theorem.
For the robust optimal control problem (34), the optimal excessof-loss reinsurance strategy is given as follows.
(ii) If 1 /( + V) < 1, we have 0 > . Thus for 0 ≤ ≤ < 0 , the inequality ≥ holds. The optimal problem is similar to that of 0 ≤ < 0 in case (i). At this time, we obtain the optimal reinsurance strategy for 0 ≤ ≤ is (68). This ends the proof of Theorem 2.
In order to get the expressions for * and the value function ( , , , ), we have to derive the expressions of ( ) and ( , ) in (57). By (58), we can rewrite (57) as Now we discuss this problem in two cases as follows.
Case 1. If 1 /( + V) ≥ 1, for 0 ≤ ≤ , the optimal reinsurance strategy for the problem (34) is (67). Denoting by 1 the function in (73), we have For equation (74), we conjecture a solution of the following form That is where with the boundary condition 1 ( ) = 0. According to the arbitrariness of l, decomposing (77) into According to [41], taking the boundary condition 1 ( ) = 0, and Similarly, for 0 ≤ < 0 , the optimal excess-of-loss strategy is (68). Denote the function by 2 in (73), and let Similar to (77)-(82), we obtain Hence Since ( , , , ) is continuous at = 0 , we have Discrete Dynamics in Nature and Society 11 and we derive  (90), it is not difficult to find that 1 ( ) = 2 ( ). Moreover, since 1 ( 0 ) = 2 ( 0 ), we obtain As a result, (87) can be rewritten as By (48), (75), and (84), we can see that the optimal investment strategy * for 0 ≤ < 0 is the same as that for 0 ≤ ≤ , which is Case 2. If 1 /( +V) < 1, we have 0 > . For 0 ≤ ≤ < 0 , the optimal excess-of-loss strategy is (68). Denoting by 3 the function in (73), we have with boundary condition ( ) = 0, 3 ( , ) = 0. Similar to the analysis for 0 ≤ ≤ in Case 1, we conjecture a solution to (94) of the following form Then employing the same method to solve the optimal problem as Case 1, a direct calculation yields 3 ( ) = 1 ( ), and Thus, for 0 ≤ ≤ , we can obtain the expression of optimal investment strategy which is the same as (93) and the corresponding value function.
The following theorem summarizes the above analysis. (100) Remark 4. Form Theorems 2 and 3, we find that (1) in the light of assumptions (40) and (41), the robust optimal strategy * depends on the parameters ℎ, , 1 , and 2 .

Verification Theorem
In this section, we will verify the candidate optimal strategies * fl * ( ) = ( * 1 ( ), * 2 ( ), * ( )) and * given by (45), Theorems 2 and 3 are indeed optimal, and the candidate value function (42) is just the value function ( , , , ) defined in (34). The main theorem is summarized as follows. Proof. It is similar to the proof of Theorem 3.2 in [42]. So we omit it here.
Next, conditions (1)-(5) in Theorem 5 will be checked. We first give two lemmas, which are useful for verifying Theorem 8.

Lemma 6. If the parameters satisfy
Proof. See Appendix.

Theorem 8. For problem (34), if the parameters satisfy the conditions in
Proof. By the proofs of Theorems 2 and 3, conditions (1)-(4) in Theorem 5 hold for ( , , , ). By Lemma 7, we know that condition (5) in Theorem 5 also holds for ( , , , ). Then, according to Theorem 5, the results of Theorem 8 are obtained.

Special Cases
In this section, we consider the robust optimal problem (34) without insurance (investment-only case), with ambiguityneutral insurer (ANI) and without delay, respectively. Since they are all special cases of (34), we only provide the results here without giving the proofs.

Corollary 10.
For the investment-only problem with delay, if 2 = = 0, 1 ( ) = 1 , the robust optimal investment strategy is (108), and the optimal value function is wherẽ1 Remark 11. According to Theorems 3 and 9 and Corollary 10, we find that the robust optimal investment strategy * ( ) of investment-only case is the same as that of reinsuranceinvestment case, which implies that the robust optimal reinsurance strategy and the robust optimal investment strategy can be separated.
In what follows, some special cases of the investment-only problem without delay are provided.
Furthermore, if 2 = = 0 in Theorem 13, the optimal strategy here along with the optimal value function will coincide with Theorem 3.1 in [23]; i.e., our model extends the results in [23] to the case of robust optimal formulation under dependent risks.

Conclusion
For the optimal control problems in insurance, most papers only consider the control systems without delay, while this paper studies a robust optimal reinsurance-investment problem with delay and dependent risks when the risky asset's price is described by Heston model. To make the optimal control problem closer to reality, we furthermore consider some possible extensions of this paper. For example, we can consider the robust equilibrium reinsurance-investment problem for a mean-variance insurer with other kinds of dependent risks, such as copulas, which is a very challenging problem.
According to (84), it is true that 1 ( ) < 0 for ̸ = ±1. In addition, for = 1, is obtained, which is According to Cauchy-Schwarz inequality, we obtain the first estimate and the last one can be derived by (A.16) and Lemma 6 under condition (101). Thus, the second part of this lemma holds.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.